### Impacto

Fernando Galván, José Francisco and Gamboa, J.M. and Ueno, Carlos
(2011)
*On convex polyhedra as regular images of R(n).*
Proceedings of the London Mathematical Society , 103
.
pp. 847-878.
ISSN 0024-6115

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Official URL: http://plms.oxfordjournals.org/content/103/5/847.full.pdf+html

## Abstract

We show that convex polyhedra in R(n) and their interiors are images of regular maps R(n) -> R(n). As a main ingredient in the proof, given an n-dimensional, bounded, convex polyhedron K subset of R(n) and a point p is an element of R(n) \ K, we construct a semialgebraic partition {A, B, T} of the boundary partial derivative K of K determined by p, and compatible with the interiors of the faces of K, such that A and B are semialgebraically homeomorphic to an (n - 1)-dimensional open ball and J is semialgebraically homeomorphic to an (n - 2)-dimensional sphere. Finally, we also prove that closed balls in R n and their interiors are images of regular maps R(n) -> R(n).

Item Type: | Article |
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Subjects: | Sciences > Mathematics > Algebraic geometry |

ID Code: | 15062 |

References: | M. Berger, Geometry. I, universitext (Springer,Berlin,1987). M. Berger, Geometry. II, universitext Springer,Berlin,1987). J. Bochnak, M. Coste and M. F. Roy,Real algebraic geometry, Ergebnisse der Mathematik 36Springer,Berlin,1998). M. Brown, ‘A proof of the generalized Schoenflies theorem’, Bull. Amer. Math. Soc. 66 (1960) 74–76. J. F. Fernando and J. M. Gamboa, ‘Polynomial images of Rn’, J. Pure Appl. Algebra 179 (2003) 241–254. J. F. Fernando and J. M. Gamboa, ‘Polynomial and regular images of Rn’, Israel J. Math. 153 (2006)61–92. G. Stengle, ‘A Nullstellensatz and a Positivstellensatz in semialgebraic geometry’, Math. Ann. 207 (1974)87–97. 8. C. Ueno, ‘On convex polygons and their complementaries as images of regular and polynomial maps of R2’,Preprint, RAAG, Fuerteventura: 2009. |

Deposited On: | 03 May 2012 09:16 |

Last Modified: | 06 Feb 2014 10:15 |

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